The present invention relates to process monitoring, and more specifically to real-time monitoring of processes wherein quality of an end product must be kept within certain desired parameters. More particularly, the present invention provides a control chart means for monitoring product quality wherein control limits are based on jackknife histograms, wherein the control limits for the range reflect a true distribution of data collected rather than a fictional normal distribution.
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It is known to use control charts in manufacturing processes to monitor the equipment used therein to ensure that the end products are produced in a consistent manner. If, for example, a physical dimension of a product exceeds a predetermined parameter, it is necessary to discontinue production, repair or recalibrate the equipment, and restart the process. Conventional control charts for size parameters (for example, X-bar charts for control limits based on the mean) are effective for their intended purpose. However, conventional control charts for variability, which rely heavily on normality assumptions which are often violated in practice, are inaccurate and may result in unnecessary stoppage of machinery to correct a nonexistent problem.
Conventional control charts for the range assume a normal distribution of data (i.e. set controls limits based on normality), and calculate parameters using a constant based on the distribution of the range. Thus, conventional control charts are symmetric, which is disadvantageous when the data obtained exhibit a skewed distribution. Disadvantageously, for equal subsample sizes, conventional (normal theory) control charts are the only control charts provided by many statistical computing packages. Accordingly, when the data distribution is not normal, conventional control charts perform poorly in assessing product parameters for quality control.
A need is therefore identified in the art for methods for monitoring variability of a product being produced by a particular piece of machinery, of of a process conducted by the machinery, or of a product stream generated thereby, which is capable of accurately detecting changes in variability in real time. The method should provide useful results in the absence of an assumption of normal data, and should establish new control limits for the range which reflect the true distribution of the data. It is further important that the method achieve the desired results without requiring an inordinate number of computations.
In accordance with a first aspect of the invention, a method for establishing a control limit for variability of a manufacturing process parameter is provided, comprising the steps of measuring the parameter over a predetermined period of time, assembling a dataset having a plurality of datapoints, wherein each datapoint represents an individual measured value of the parameter, rank-ordering the datapoints within the dataset, and selecting at least one subset of the dataset, wherein the subset is a predetermined number of datapoints having a high measured value and a low measured value defining a rank-ordered range of measured parameter values. All possible numbers of subsamples of the dataset having the range defined by the selected subset are calculated in accordance with formulae which will be described below. The steps of selecting a subset and calculating all possible numbers of subsamples of the dataset having the range defined by the selected subset are repeated until all possible numbers of ranges have been calculated for all possible subsets of the predetermined size to define a set of possible ranges. The set of possible ranges is then rank-ordered.
Finally, control limits for the range for the manufacturing process are established, defined by an upper control limit and a lower control limit, wherein the upper and lower limits are defined as a predetermined percentile of the rank-ordered set of ranges. The method of the present invention may be adapted to a jackknife method (without sample replacement) or a bootstrap method (with sample replacement) for computing control limits.
For the jackknife method, if the ordered data set from smallest to largest is denoted x(1), x(2), . . . x(N) and the subsample size is n, then all possible numbers of a datapoint subsample having the range defined by the subset (x(h)xe2x88x92x(g) are calculated in accordance with the formula:       (                                        h            -            g            -            1                                                            n            -            2                                )    "AutoLeftMatch"
where h is the highest measured value within a range, g is the lowest measured value within a range, and n is the number of observations within the subset.
For the bootstrap method, the calculation relies on the relationship between the highest measured value within a range and the lowest measured value within a range. When the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g=h where h is the highest measured value within a range and g is the lowest measured value within a range, the range is zero for n subsamples. When the relationship between g and h is defined by the equation g=hxe2x88x921, the control limits are defined by the range of 2nxe2x88x922 subsets, where n is the number of observations within the subset. When the relationship between the highest measured value and the lowest measured value in a range is defined by the equation g less than hxe2x88x921, all possible numbers of a subsample of the dataset having the range defined by the subset (depicted as x(h)xe2x88x92x(g)) are calculated in accordance with the formula:       ∑          c      =      1              n      -      1        ⁢            ∑              d        =        1                    n        -        c              ⁢                  (                                            n                                                          c                                      )            ⁢              (                                                            n                -                c                                                                        c                                      )            ⁢                        (                      h            -            g            -            c            -            d            +            1                    )                          n          -          c          -          d                    
where n is the number of observations within the subset, c is the number of times the lowest measured value appears in the subset, d is the number of times the highest measured value appears in the subset, h is the highest measured value within a range and g is the lowest measured value within a range.
In another aspect, the present invention provides a computer software program for establishing a control limit for variability of a manufacturing process parameter, wherein the software performs the steps of rank-ordering a plurality of datapoints within a dataset, wherein each datapoint represents an individual measured value of the parameter. Next the software selects at least one subset of the dataset, wherein the subset is a predetermined number of datapoints having a high measured value and a low measured value defining a rank-ordered range of measured parameter values. All possible numbers of subsamples of the dataset having the range defined by the selected subset are calculated by the software in accordance with formulae described above. The software then repeats the steps of selecting a subset and calculating all possible numbers of subsamples of the dataset having the range defined by the selected subset until all possible numbers of ranges have been calculated for all possible subsets of the predetermined size to define a set of possible ranges. The set of possible ranges is then rank-ordered.
The software then sets control limits for the range for the manufacturing process, defined by an upper control limit and a lower control limit, wherein the upper and lower limits are defined as a predetermined percentile of the rank-ordered set of ranges. Finally, the software performs the step of comparing subsequent datapoints against the control limits to ensure that the parameter is within a predetermined variability range. The software may be used to establish jackknife and bootstrap control limits as described above, and in accordance with the equations described above.
In yet another aspect of the invention, a method is provided for establishing a control limit for variability of a manufacturing process parameter, comprising the steps of measuring the parameter over a predetermined period of time, and assembling a dataset having a plurality of datapoints representing a measured value of the parameter. Next, for each possible value for the range of a predetermined subsample size, the number of subsamples within the dataset having that range is calculated using the formulae described above. The set of ranges derived thereby is rank-ordered, and control limits for the manufacturing process are established as upper and lower control limits defined as a percentile of the rank-ordered set of ranges.